On 2D Inverse Problems/On the inverse problem of Calderon

The following inverse problem stated by Calderon has many potential practical applications and had received a lot of attention in the past decades, see [Uh]. This is the problem of recovering the conductivity of a body from its Dirichlet-to-Neumann operator.

Given a domain with positive measurable function on it, the Dirichlet-to-Neumann operator connects the Dirichlet and Neumann boundary values of $\gamma$-harmonic functions in $H^{1}(\Omega )$, defined on the domain. It is a pseudo-differential operator of order 1.

$\Lambda :H^{1/2}(\partial \Omega )\rightarrow H^{-1/2}(\partial \Omega )$

One can only recover one number (an integral of reciprocal of conductivity) in 1D. The problem is overdetermined in dimensions higher than 2. It was settled in these cases @ [KV] & [SU].

The dimensions of the measurement parameter and the unknown one fit precisely in the case of 2D. It was recently proved that the Dirichlet-to-Neumann map uniquely determines the conductivity in a 2D simply connected bounded domain if the conductivity is in a weighted space including differentiable functions in [N], and if it's measurable and bounded from 0 and infinity in [AP].

$0<1/c<\gamma <c<\infty .$

In the 2D case one can make sense of the operator for measurable conductivities (vs. differentiable), (see [AP]), using the Hilbert transform, defined later in the book, b/w the boundary values of harmonic functions and their conjugates.

In this book the problem is also stated for the planar electrical networks, that allows a different approach for solution of the inverse problem through discretization technique, that was implemented by Druskin et al. @ Schlumberger-Doll Research, see [BDK] and [BDMV].